A street is a long thing. Geometrically, it's a line and not a point. So it is not usually a destination in and of itself (although the intersection of two streets could be a destination). The person's destination is probably a particular address on/in that street (notice that we say "on" or "in" in English, and not "at"). It's also possible that the person will simply pass down that street on the way to his final destination.

If the person were in fact going to the same building, he might reply Vous pouvez me suivre ; je vais à la même adresse. But when he says Je vais dans la même rue, he just means that he will be in the street in question on his way to his destination.